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A054493 A Pellian-related recursive sequence.

%I A054493 #63 Jul 02 2025 16:01:59
%S A054493 1,7,36,175,841,4032,19321,92575,443556,2125207,10182481,48787200,
%T A054493 233753521,1119980407,5366148516,25710762175,123187662361,
%U A054493 590227549632,2827950085801,13549522879375,64919664311076,311048798676007,1490324329068961,7140572846668800
%N A054493 A Pellian-related recursive sequence.
%C A054493 This is the r=7 member in the r-family of sequences S_r(n+1) defined in A092184 where more information can be found.
%C A054493 Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 7, P2 = 10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 25 2014
%D A054493 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H A054493 G. C. Greubel, <a href="/A054493/b054493.txt">Table of n, a(n) for n = 0..1000</a>
%H A054493 Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%H A054493 I. Adler, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), 181-193.
%H A054493 Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>
%H A054493 S. Barbero, U. Cerruti, and N. Murru, <a href="http://www.seminariomatematico.polito.it/rendiconti/78-1/BarberoCerrutiMurru.pdf">On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy</a>, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
%H A054493 E. I. Emerson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%H A054493 Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.
%H A054493 R. Stephan, <a href="http://www.ark.in-berlin.de/A001110.ps">Boring proof of a nonlinearity</a>
%H A054493 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A054493 H. C. Williams and R. K. Guy, <a href="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume
%H A054493 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A054493 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6,1)
%F A054493 a(n) = 5*a(n-1) - a(n-2) + 2, a(0)=1, a(1)=7.
%F A054493 A004254 = sqrt{21*(A054493)^2+28*(A054493)}/7. - _James Sellers_, May 10 2000
%F A054493 a(n) = (1/3)*(-2 + ((5+sqrt(21))/2)^n + ((5-sqrt(21))/2)^n). - _Ralf Stephan_, Apr 14 2004
%F A054493 G.f.: (1+x)/((1-x)*(1 - 5*x + x^2)) = (1+x)/(1 - 6*x + 6*x^2 - x^3). From the R. Stephan link.
%F A054493 a(n) = 6*a(n-1) - 6*a(n-2) + a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=7.
%F A054493 a(n) = (2*T(n, 5/2)-2)/3, with twice the Chebyshev polynomials of the first kind, 2*T(n, x=5/2)=A003501(n).
%F A054493 a(n) = b(n) + b(n-1), n>=1, with b(n)=A089817(n) the partial sums of S(n, 5)= U(n, 5/2)=A004254(n+1), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind.
%F A054493 From _Peter Bala_, Mar 25 2014: (Start)
%F A054493 The following formulas assume an offset of 1.
%F A054493 Let {u(n)} be the Lucas sequence in the quadratic integer ring Z[sqrt(7)] defined by the recurrence u(0) = 0, u(1) = 1 and u(n) = sqrt(7)*u(n-1) - u(n-2) for n >= 2. Then a(n) = u(n)^2.
%F A054493 Equivalently, a(n) = U(n-1,sqrt(7)/2)^2, where U(n,x) denotes the Chebyshev polynomial of the second kind.
%F A054493 a(n) = 1/3*( ((sqrt(7) + sqrt(3))/2)^n - ((sqrt(7) - sqrt(3))/2)^n )^2.
%F A054493 a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -5/2; 1, 7/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
%F A054493 See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
%F A054493 a(2*n - 1) = 7 * A004254(n)^2, a(2*n) = A030221(n)^2 for all n in Z. - _Michael Somos_, Jan 22 2017
%F A054493 a(n) = a(-2-n) for all n in Z. - _Michael Somos_, Jan 22 2017
%F A054493 0 = 1 + a(n)*(-2 + a(n) - 5*a(n+1)) + a(n+1)*(-2 + a(n+1)) for all n in Z. - _Michael Somos_, Jan 22 2017
%e A054493 G.f. = 1 + 7*x + 36*x^2 + 175*x^3 + 841*x^4 + 4032*x^5 + 19321*x^6 + ...
%p A054493 A054493 := proc(n)
%p A054493     option remember;
%p A054493     if n <= 1 then
%p A054493         6*n+1 ;
%p A054493     else
%p A054493         5*procname(n-1)-procname(n-2)+2 ;
%p A054493     end if ;
%p A054493 end proc:
%p A054493 seq(A054493(n),n=0..10) ; # _R. J. Mathar_, Apr 16 2018
%t A054493 LinearRecurrence[{6,-6,1},{1,7,36},30] (* _Harvey P. Dale_, Apr 15 2015 *)
%t A054493 a[ n_] := ChebyshevU[n, Sqrt[7]/2]^2; (* _Michael Somos_, Jan 22 2017 *)
%o A054493 (PARI) {a(n) = simplify(polchebyshev(n, 2, quadgen(28)/2)^2)}; /* _Michael Somos_, Jan 22 2017 */
%Y A054493 Cf. A004254, A100047, A030221 (first differences).
%K A054493 easy,nonn
%O A054493 0,2
%A A054493 _Barry E. Williams_, May 06 2000
%E A054493 Chebyshev comments from _Wolfdieter Lang_, Sep 10 2004